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Approximation of functions by linear means of their Fourier series
Nonlinear
Analysis,
Theory,
Methods
Pergamon PII: SO362-546X(a7)00225-3
&Applications. Vol. 30, No. 4, pp. 2S39-2540. 1991 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
APPROXIMATION OF FUNCTIONS LINEAR MEANS OF THEIR FOURIER YURI Department Key
words
Abstract.
The
of Fourier both
and
special
series
above
and
of Mathematics, phrases:
representations
of these below
Let a function
smoothness
functions
by modulus
f E L,t ,
L. NOSENKO
Done&k
State
modulus,
arithmetical
obtained.
The
of smoothness
-
expansion
for
of given
2
qeikx
k=-m
for f(z).
University,
means,
integrable
appropriate
1 5 p 5 co, 2r-periodic f(x)
the Fourier
Technical
of a deviation are
BY SERIES
Ukraine
multiplicators,
functions remainders
deviation
and are
linear
means
estimated
from
functions.
and
=
2
Ak(x)
(1)
-
k=-co
L e t us consider
arithmetical
means
for (1) i.e.
~n(f;x) =,k,~n(Gi)~*(x)c Let A and w are symmetric difference and modulus of smoothness and steps) respectively. The deviation f(z) - a,(f; x ) was investigated by some authors Here is one of these results due to Lebed’ H.K. and Avdienko A.A.
(of appropriate in different [l]:
orders directions.
f(x) - a,(f; x) = -& lrn A:,tn+~~f(+-2~~+ ~n(f;x)> IlGt(f;x)llp I c4.f; l/(n + l))p’r c > 0. improper integrals of difIn [2] we have represented f(z) - a,(f; z ) as a sum of appropriate ferences for f and conjugated function of second-forth orders and with a remainder estimated from above and below by a fourth order modulus of smoothness for f 3 Cj > 0,
f(x)
- an(f;
x) = C; lrn
A;,~n+~~f(~)~-~~~
+ C; Jim + c;
(1 Present
address:
Donetsk
State
Technical
University, 2539
Artyoma
&,~n+qf”bW3~~+ A;,C,+l,f(+-4dt
r
+ ~r~(f; x),
1
str.
58, Donetsk,
340000,
Ukraine
Second World
2540 C1W4(f;
l/(n
Now we give a new result THEOREM.
3
Cl
Let
>
0,
f E L,,
CT2 >
+ 1))~
in this
Congress
of Nonlinear
5 llTn(f;
x)llp
:
f(x)
-
u,(f;x)
M
/cn
+ l))p.
with
the Fourier
expansion
(1).
zz
+
A~lj:n:l~f(,)t-(2j+1)dt
ll(n + l)jp 5 Il~~(f; l/(n + l)jp I ll~n(f; x)llp I
+ ~~(f; z),
C2w2(.
. .I,
The idea of the proof is the same as in [2]: the p rinciple of comparison R.M. [3] and theorems on multiplicators. The results which are like to given one in the theorem above are obtained Riesz means (CY > 0)
And
arithmetic
Then
CD
A:;(n+l)f(“)t-23d~ C1w2~(f;
5 C2“J4(f;
direction.
1 < p 5 00, 2n-periodic
0 :
Analysts
m E N.
due to Trigub in
the
caSe
of
means. REFERENCES
1. LEBED’
H.K., AVDIENKO A.A., Concerning the approximation of Iz~ AN SSSR ser mal. v.35 (1971), 83 - 92. (Russian) 2. NOSENKO YU.L., Approximation of functions by Riesz, (C, a) and these functions, Bul. St. al Univ. Baia Mare, ser. B vol. X (1994), 3. TRIGUB R.M., Absolute convergence of Fourier integrals, Izv. AN 1409. (Russian)
periodical typical 89-92. SSSR,
functions means ser.
for
mat.
by Fejer Fourier
44 (1980),
sums,
series
of
1379%
Analysis,
Theory,
Methods
Pergamon PII: SO362-546X(a7)00225-3
&Applications. Vol. 30, No. 4, pp. 2S39-2540. 1991 Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Ekvier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
APPROXIMATION OF FUNCTIONS LINEAR MEANS OF THEIR FOURIER YURI Department Key
words
Abstract.
The
of Fourier both
and
special
series
above
and
of Mathematics, phrases:
representations
of these below
Let a function
smoothness
functions
by modulus
f E L,t ,
L. NOSENKO
Done&k
State
modulus,
arithmetical
obtained.
The
of smoothness
-
expansion
for
of given
2
qeikx
k=-m
for f(z).
University,
means,
integrable
appropriate
1 5 p 5 co, 2r-periodic f(x)
the Fourier
Technical
of a deviation are
BY SERIES
Ukraine
multiplicators,
functions remainders
deviation
and are
linear
means
estimated
from
functions.
and
=
2
Ak(x)
(1)
-
k=-co
L e t us consider
arithmetical
means
for (1) i.e.
~n(f;x) =,k,~n(Gi)~*(x)c Let A and w are symmetric difference and modulus of smoothness and steps) respectively. The deviation f(z) - a,(f; x ) was investigated by some authors Here is one of these results due to Lebed’ H.K. and Avdienko A.A.
(of appropriate in different [l]:
orders directions.
f(x) - a,(f; x) = -& lrn A:,tn+~~f(+-2~~+ ~n(f;x)> IlGt(f;x)llp I c4.f; l/(n + l))p’r c > 0. improper integrals of difIn [2] we have represented f(z) - a,(f; z ) as a sum of appropriate ferences for f and conjugated function of second-forth orders and with a remainder estimated from above and below by a fourth order modulus of smoothness for f 3 Cj > 0,
f(x)
- an(f;
x) = C; lrn
A;,~n+~~f(~)~-~~~
+ C; Jim + c;
(1 Present
address:
Donetsk
State
Technical
University, 2539
Artyoma
&,~n+qf”bW3~~+ A;,C,+l,f(+-4dt
r
+ ~r~(f; x),
1
str.
58, Donetsk,
340000,
Ukraine
Second World
2540 C1W4(f;
l/(n
Now we give a new result THEOREM.
3
Cl
Let
>
0,
f E L,,
CT2 >
+ 1))~
in this
Congress
of Nonlinear
5 llTn(f;
x)llp
:
f(x)
-
u,(f;x)
M
/cn
+ l))p.
with
the Fourier
expansion
(1).
zz
+
A~lj:n:l~f(,)t-(2j+1)dt
ll(n + l)jp 5 Il~~(f; l/(n + l)jp I ll~n(f; x)llp I
+ ~~(f; z),
C2w2(.
. .I,
The idea of the proof is the same as in [2]: the p rinciple of comparison R.M. [3] and theorems on multiplicators. The results which are like to given one in the theorem above are obtained Riesz means (CY > 0)
And
arithmetic
Then
CD
A:;(n+l)f(“)t-23d~ C1w2~(f;
5 C2“J4(f;
direction.
1 < p 5 00, 2n-periodic
0 :
Analysts
m E N.
due to Trigub in
the
caSe
of
means. REFERENCES
1. LEBED’
H.K., AVDIENKO A.A., Concerning the approximation of Iz~ AN SSSR ser mal. v.35 (1971), 83 - 92. (Russian) 2. NOSENKO YU.L., Approximation of functions by Riesz, (C, a) and these functions, Bul. St. al Univ. Baia Mare, ser. B vol. X (1994), 3. TRIGUB R.M., Absolute convergence of Fourier integrals, Izv. AN 1409. (Russian)
periodical typical 89-92. SSSR,
functions means ser.
for
mat.
by Fejer Fourier
44 (1980),
sums,
series
of
1379%